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In the summary of [Leshchenko, Yu. Yu. The structure of an infinite wreath power construction of the regular group of prime order p. (Ukrainian. English summary) Zbl 1164.20335 Mat. Stud. 28, No. 2, 141-146 (2007)] it is mentioned that

"The structure of an infinite wreath power construction of the regular group of prime order p is considered. It is shown that such infinite wreath power construction is verbally complete."

Since I cannot read the Ukrainian content, I can just think only on the summary. If this result were true, then every element of $G:=C_p\wr C_p\dots\wr C_p\wr\dots$ would be a commutator whenever we choose $w(x_1,x_2)=[x_1,x_2]$ (a commutator) and then $G$ would be perfect, i.e. $G'=G$. But it is known that $G$ is not perfect. Am I missing something? (See also [Yu. Yu. Leshchenko, Fully invariant subgroups of an infinitely iterated wreath product, Algebra and Discrete Mathematics, 12 (2) (2011) 85-93.]

Ps: A group $G$ is called verbally complete if for an arbitrary $g \in G$ and for an arbitrary non-trivial word $w(x_1, x_2, \dots , x_n)$ there are $g_1, g_2, \dots, g_n \in G$ such that $w(g_1, g_2, \dots, g_n) = g$).

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  • $\begingroup$ At MSN the title is translated as "Construction of an infinite iterated wreath power of a regular cyclic group of prime order $p$". The review (by Yu. Bornachuk): "A construction of a truncated wreath product of an infinite sequence of permutation groups is introduced. In particular, the inductive limit $U^\infty_p$ of wreath powers of a cyclic prime order group, which is constructed by the diagonal embedding, can be obtained by it. A main result of the paper is that the group $U^\infty_p$ is verbally complete. (...) $\endgroup$
    – YCor
    Commented Aug 16, 2022 at 14:04
  • $\begingroup$ (...) Recall that the verbal completeness of a group $G$ means that for any nontrivial word $w=w(x_1,\dots,x_n)$ and element $g\in G$ there are elements $g_1,…,g_k\in G$ such that $w(g_1,…,g_n)=g$. The proof of the verbal completeness of $U^\infty_p$ is deduced from a result of P. Hall about locally finite groups." $\endgroup$
    – YCor
    Commented Aug 16, 2022 at 14:04
  • $\begingroup$ My guess is that you're not using the same sequence of embeddings as the author. Namely when embedding $A$ into $A\wr C_p=A^p\rtimes C_p$, the author embeds $A$ not as a single factor, but uses the diagonal embedding (into $A^p$). Indeed, assuming that the abelianization of $A$ has exponent $p$, this embeds $A$ into the derived subgroup of $A\wr C_p$. And the abelianization of the $n$-fold iterated wreath product of $C_p$ is $C_p^n$. So the inductive limit is indeed perfect. $\endgroup$
    – YCor
    Commented Aug 16, 2022 at 14:16
  • $\begingroup$ Actually the author explains the details in the second paper that I cited. I should read the paper more carefully. It may have different meaning other than in my mind. Thank you. $\endgroup$
    – IGT
    Commented Aug 16, 2022 at 15:03

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